"I am unwavering in my dedication to the B.C. Conservatives, and fully intend to lead my party into the next general election," he said in a written release.
"Those few but vocal party members who are critical of my leadership and do not support me have until … noon Wednesday to either get on-side or quit our party and join another."
Cummins also said the party would refund membership fees and return financial donations made in the past year to anyone who quits the party by the deadline.
"Since I became leader of the B.C. Conservatives in the spring of 2011, our party has risen substantially in public opinion polls, and we have recruited outstanding individuals who will represent the party in the May 14, 2013 general election,” Cummins said in the release.
"I foresee no circumstances under which I would not be leading the B.C. Conservatives to victory next May."
Last month, John van Dongen — who was the party's only sitting MLA — resigned from the party after members voted against holding a leadership review. The dispute centred around Cummins’s leadership style and his $4,000 a month stipend.
The push to oust Cummins, a former Conservative MP federally, was led by the party's vice-president Ben Besler.
The most recent poll by Angus Reid put the Conservatives in third place with 19 per cent voter support, behind the governing Liberals at 25 per cent and the opposition NDP at 46 per cent.